This looks typical of fossilised coral to me. There is a lot of this around the red sea resort of Sharm el Shiek where an earth quake caused the land to rise a few thoasand years ago and left the coral reefs high and dry. I have also seen similar coral in the UK but from Millions of years ago.

It must surely be coral, but being on pottery (even if a few 1,000 yr old), surely not actually fossilised? Given that you say it's on a beach (I assume a modern one), then if the species is around today it shouldn't take long.

Rate of growth is affected by several factors: water termperature and motion, light levels, nutrient levels, symbiosis, the particular coral species, etc.

In favourable conditions, coral can grow 20cm in a year

How old is your particular piece of coral? Well it is younger than the piece of pottery, sure. But it is probably a matter of a few years. You might be able to see periodic (perhaps annual) growth lines.

Thanks to all of you, John ,rhossilian and especially to you Mike, for the detailed and helpful answer including references that were so helpful.

Your answers convinced me that I did find a coral growing on a piece of pottery in the eastern coast of the mediterranean. growing on such a small shred means it had to grow pretty fast as suvh a piece will not lie steady for very long time. I did not realize there are such favorable conditions there.

Just to make sure I went yesterday to Eilat beach (on the Red sea) and took the attached picture of the original piece with a coral swept to the beach of Eilat